#460416

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1314 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6,9,8)(7,11,12,10)(13,14,19,24)(15,25,30,43)(16,21,32,40)(17,33,31,22)(18,34,26,23)(20,36,37,29)(27,41,44,38)(28,42,35,39);(5,7,9,12)(6,10,8,11)(13,15,26,36,19,30,18,29)(14,20,34,43,24,37,23,25)(16,31,35,27,32,17,28,44)(21,38,42,33,40,41,39,22);(2,4)(6,8)(10,11)(13,16)(14,21)(15,27)(17,29)(18,35)(19,32)(20,33)(22,37)(23,42)(24,40)(25,41)(26,28)(30,44)(31,36)(34,39)(38,43);(5,8,9,6)(7,11,12,10)(13,17,18,27,19,31,26,44)(14,22,23,41,24,33,34,38)(15,28,29,32,30,35,36,16)(20,21,25,39,37,40,43,42)

Polycyclic presentation:

2^9|a^2=eg,b^2=ei,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,g^c=gi,h^c=hi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 367612);